For this project, I modeled a two-mass system in the 251 Laboratory, which utilizes a rectilinear system. Previously, the system was modeled with one mass employed and used for laboratory experiments for students enrolled in ESE 251. The system (detailed in the following section titled “System Description”) represents the dynamics found in many mechanical systems. Each mass can have two modes, sine or cosine. I will conclude what kind of sinusoidal inputs will be needed to induce the two modes. First, I conduct experiments with the system. Then, I model an ideal version of the two-mass system. I then formulate the space-state representation and the differential equations from this model that represent the system. I use the A matrix and various Matlab techniques to obtain the eigenvalues, modes, and frequency response. The outcomes of this project will be as follows: (1) the models and equations describing the system, (2) the inputs which invoke certain modes, and (3) a new laboratory experiment with solution guide for ESE 251.
It is known that an n-degree
of freedom system will have n natural
frequencies and n natural modes of
vibration. In this case n=2 because there are two masses, and
hence I expect that there will appear to be two natural frequencies. I set out to answer the following
questions: How do we find these natural
frequencies? What do they mean? Why are they important? I expect the system to exhibit a combination
of two kinds of behavior (modes) depending on the input. One behavior occurs when Mass1 moves in the
positive or negative direction the same time that Mass2 is moves in the same
direction. The second type of behavior
occurs when Mass1 moves in the opposite direction that Mass2 moves. We will call these behaviors “in-phase” and
“out-of-phase” modes, respectively.
MODELING AN IDEALIZED TWO-MASS/SPRING SYSTEM
MODELING AN IDEALIZED TWO-MASS/SPRING SYSTEM WITH DAMPERS
PARAMETER
IDENTIFICATION OF A SINGLE-MASS/SPRING/DAMPER SYSTEM
PARAMETER IDENTIFICATION OF A TWO-MASS/SPRING/DAMPER SYSTEM
The natural frequency is a crucial
aspect of a system. When an oscillatory
external force is applied, the system is forced to vibrate at that excitation
frequency. If the applied frequency is
equal to or close to the natural frequency of the system, large or unstable
oscillations can occur, known as resonance.
This concept is important when building any major structure, such as
bridges, buildings, or airplane wings.
The most famous example is the
In the two-mass setup of the ECP Model 210, I was able to excite the modes so that the masses traveled in-phase by applying a sinusoidal input with a frequency lower than the natural frequency of the system. Once I reached close to 5Hz, the system oscillations were very large and growing and hence exhibiting resonance. At higher input frequencies I was able to excite the modes so that the masses traveled out-of-phase with each other.
I was able to accurately model an ideal version of the actual system with and without dampers incorporated. An analysis of the A matrix and the transfer functions helped determine the modes, as well as the natural frequency. The damper does not greatly effect the location of the natural frequencies of the system but does reduce the magnitude of oscillations possible when the input frequency matches the natural frequency.
Through the labs that I created, I guide students through understanding numerous techniques to help model and understand systems, such as eigenvalues, poles, mode functions, and frequency response. I explored all of these in my research as well. Hopefully the laboratory exercises will develop their skills in modeling systems, understanding the importance of frequency response by using bode diagrams and system response graphs, and analyzing equations to obtain and utilize eigenvalues and modes. They will also gain a better understanding of parameter identification and natural frequencies.
ESE 251
LABORATORY SOLUTION GUIDE
Educational Control
Products.
<http://www.ecpsystems.com/>
Mukai, Hiro. Introduction to Mechanical
and Electrical Systems. Chapters 1-9. ESE 351 (Signals and Systems) Class Notes
"Linear Systems and Signals," Dept of Electrical and Systems
Engineering.
Mukai, Hiro. Laboratory for ESE 251 (Introduction to Systems Science
and Engineering). Laboratories 1-12, Dept of Electrical and Systems
Engineering.